Applying the brake on the slipping wheel is one half of traction control. Reducing the power going to the wheel is the other half. I accidentally demonstrated this to myself many years ago when my car suddenly was almost completely ignoring my throttle input for a reason I couldn't explain. It turned out I still had the parking brake on and when I went over a patch of ice and the car lost traction, adding more brake wouldn't do much, so it simply ignored my throttle. Because I wasn't moving more than about 5 miles per hour to begin with, it felt like the car just refused to move, like a stubborn mule.
It is amazingly effective indeed. One time I was driving through ice and snow, and at the traffic light I turned of the ESP by mistake (was trying to plug in my phone charger in the lighter socket beside the button). When I took off I nearly crashed myself off the road, I never realised how slippery it was, and how much the car was covering for my ignorance.
At least this is how it was explained to me by my mechanic friend